Journal of Mathematics
Volume 2020 (2020), Article ID 9651894, 5 pages
https://doi.org/10.1155/2020/9651894

Research Article

-Coherence Relative to a Hereditary Torsion Theory

Zhu Zhanmin

Department of Mathematics, Jiaxing University, Jiaxing, Zhejiang Province 314001, China

Correspondence should be addressed to Zhu Zhanmin; zhuzhanminzjxu@hotmail.com

Received 13 January 2020; Accepted 6 July 2020; Published 4 August 2020

Academic Editor: Elena Guardo

Copyright © 2020 Zhu Zhanmin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Let be a ring, a hereditary torsion theory of mod-, and a positive integer. Then, is called right --coherent if every -presented right -module is -presented. We present some characterizations of right --coherent rings, as corollaries, and some characterizations of right -coherent rings and right -coherent rings are obtained.

1. Introduction

Throughout this paper, is an associative ring with identity and all modules considered are unitary. For any -module , will be the character module of .

Recall that a torsion theory [1] for the category of all right -modules consists of two subclasses and such that

(1) for all and (2)If for all , then (3)If for all , then

In this case, is called a torsion class and its objects are called -torsion, is called a torsion-free class, and its objects are called -torsion free. From, Proposition 2.1, Chap VI, in [9], a class of right -modules is a torsion class for some torsion theory if and only if is closed under quotient modules, direct sums, and extensions. From Proposition 2.2, Chap VI in [9], a class of right -modules is a torsion-free class for some torsion theory if and only if is closed under submodules, direct products, and extensions. A torsion theory is called hereditary if is closed under submodules.

We recall also that a right -module is called FP-injective [2] or absolutely pure [3] if for every finitely presented right -module ; a left -module is flat if and only if for every finitely presented right -module ; a ring is right coherent [4] if every finitely generated right ideal of is finitely presented, or equivalently, if every finitely generated submodule of a projective right -module is finitely presented. -injective modules, flat modules, coherent rings, and their generalizations have been studied extensively by many authors. For example, in 1994, Costa introduced the concept of right n-coherent rings in [5]. Following [5], a ring is called right -coherent in case every -presented right -module is -presented, where a right -module is called -presented in case there exists an exact sequence of right -modules , in which every is finitely generated free. It is easy to see that a ring is right coherent if and only if is right 1-coherent. In 1996, Chen and Ding introduced the concepts of n-FP-injective modules and n-flat modules in [6], using the two concepts and characterized right -coherent rings. Following [6], a right -module is called --injective in case for every -presented right -module ; a left -module is called -flat in case for every -presented right -module .

Let be a (hereditary) torsion theory for the category of all right -modules. Then, according to [7], a right -module is called right -finitely generated (or -FG for short) if there exists a finitely generated submodule such that ; a right -module is called -finitely presented (or -FP for short) if there exists an exact sequence of right -modules with finitely generated free and -finitely generated; is called -coherent if every finitely generated right ideal of is -FP. In 1993, Nieves introduced the concept of -n-presented (or -n-FP for short) modules in [8]. Let be a torsion theory for the category of all right -modules; then, according to [8], a right -module is called --presented in case there exists an exact sequence of right -modules , where each is finitely generated free and is -finitely generated. Let be a hereditary torsion theory. Then, by Theorem 3.3 in [4], it is easy to see that is right -coherent if and only if every finitely presented right -module is -2-FP.

In this article, we wish to extend the concepts of right -coherent rings and right -coherent rings to right --coherent rings and give some characterizations of these rings (see Theorems 1 and 2), as corollaries; some characterizations of right -coherent rings and right -coherent rings will be obtained (see Corollaries 2 and 4).

2. Characterizations of --Presented Modules and Right --Coherent Rings

We recall that a nonempty subclass of right -modules is called a weak torsion class [9] if is closed under homomorphic images and extensions. Following [9], if a class of right -modules is a weak torsion class, then a right -module is called -finitely generated (or -FG for short) if there exists a finitely generated submodule such that ; a right -module is called -finitely presented (or -FP for short) if there exists an exact sequence of right -modules with finitely generated free and -finitely generated; a right -module is called -presented if there exists an exact sequence of right -modules:

such that are finitely generated free and is -finitely generated, where is a positive integer. Let be a torsion theory of mod-R. Then, it is easy to see that is a weak torsion class. In this case, a right -module is -finitely generated if and only if it is -finitely generated, and a right -module is --presented if and only if it is -presented for any positive integer . Let be a right -module, a torsion theory of mod-R, and a positive integer. Then, form Proposition 3.4 in [11], the following conditions are equivalent:

(1) is --presented.(2) is -presented, and if there exists an exact sequence of right -modules:

 such that are finitely generated free; then, is -finitely generated.(3)There exists an exact sequence of right -modules:

 such that is finitely generated free and is --presented. If , then the above conditions are also equivalent to(4) is -presented, and if there exists an exact sequence of right -modules

 such that are finitely generated free; then, is -finitely presented.

Now, we give the characterizations of --presented modules.

Theorem 1. Let be a torsion theory of mod-R, a nonnegative integer, and an -presented right -module. Then, the following statements are equivalent for A:

(1) is --presented(2)The canonical map is an isomorphism for each direct system of -torsion-free modules(3) for each -torsion-free module X(4) for each -torsion-free injective module E

Proof.

  Use induction on . If , then the result holds by Proposition 2.5(3) in [4]. Assume that the result holds when . Then, when , suppose is a --presented module. Let be an exact sequence of right -modules, where is finitely generated free and is --presented. Then, we have a commutative diagram:

 with exact rows. Since is an isomorphism and hence epic and is an isomorphism by hypothesis, we have that is also an isomorphism by the Five Lemma.  Use induction on . If , then the result holds by Proposition 2.5 (3) in [4]. Assume that the result holds when . Then, when , suppose is a -presented right -module. Let be an exact sequence of right -modules, where is finitely generated free and is -presented. Then, for any direct system of -torsion-free modules. If , then we have a commutative diagram:

 with exact rows. Since is an isomorphism by condition, we have that is also an isomorphism. So, by hypothesis, is --presented, and hence is --presented. If , then we have a commutative diagram:

 with exact rows. From 25.4 (d) in [10], is an isomorphism and hence epic, and is an isomorphism by condition. Note that is an isomorphism, so, by the Five Lemma, we have that is also an isomorphism. So, is -FP by Proposition 2.5 in [4], and it shows that is -2-FP.  In case, , then the result holds by Lemma 3.1 in [4]. In case, , then there is an exact sequence of right -modules , where is finitely generated free and is -FP. And then we have a commutative diagram:

 with exact rows. By Lemma 3.1 [4], and are isomorphisms. So, by the Five Lemma, is also an isomorphism. In case, , then we have an exact sequence of right -modules , where each is finitely generated free and is -2-FP, and hence we have , as required.  It is obvious.  Since is -FP, there exists an exact sequence of right -modules , where each is finitely generated free and is finitely presented. Thus, for any -torsion-free injective module by (4). It follows from Proposition 2 in [6] that is -2-FP, and therefore is --presented.

Let be a right -module and be a positive integer. If , then it is easy to see that is -presented if and only if it is -presented. If , then it is easy to see that is -presented if and only if it is -presented.

Corollary 1. Let be a nonnegative integer and an -presented right -module. Then, the following statements are equivalent:

(1) is -presented(2)The canonical map is an isomorphism for each direct system of right R-modules(3) for each right -module X(4) for each injective right -module E

Definition 1. Let be a torsion theory of mod-R. Then, the ring is called right -n-coherent, if every n-presented right R-module is -presented.
Let be a torsion theory of mod-R. Then, it is easy to see that is right --coherent if and only if every -presented right -module is -presented.

Example 1.

(1)Let . Then, is right --coherent if and only if is right -coherent.(2) is right -coherent if and only if is right -1-coherent.(3)Let . Then, is right --coherent.

Proof. (1) and (3) are obvious. (2) follows from Theorem 3.3 (2) in [4].

Definition 2. Let be a torsion theory of mod-R and n a positive integer. Then, a right -module M is said to be -n-FP-injective, if for each --presented module M; a right -module M is said to be -FP-injective if it is -1-FP-injective.
Clearly, each -FP-injective module is --FP-injective. If , then it is easy to see that a right -module is --FP-injective if and only if it is -FP-injective. Now, we give our characterization of right --coherent rings.

Theorem 2. Let be a hereditary torsion theory of mod-R and n a positive integer. Then, the following statements are equivalent for the ring R:

(1)R is right -n-coherent(2) for any n-presented right -module A and direct system of -torsion-free modules(3) for any n-presented right -module A and each -torsion-free module X(4) for any n-presented right -module A and each -torsion-free injective module E(5)If X is a -n-FP-injective module, then X is n-FP-injective(6)Any direct limit of -torsion-free n-FP-injective modules is n-FP-injective(7)Any direct limit of -torsion-free FP-injective modules is n-FP-injective(8)Any direct limit of -torsion-free injective modules is n-FP-injective(9)A -torsion-free module X is n-FP-injective if and only if is n-flat(10)If Y is a pure submodule of a -torsion-free n-FP-injective module X, then is n-FP-injective.

Proof.

  follows from Theorem 1. , , and are obvious.  Let , where each is a -torsion-free -FP-injective module. Then, for any --FP module , by Theorem 1, we have that , so is --FP-injective and thus it is -FP-injective by (5). . Let be an -presented right -module with a finite -presentation . Write and . Then, is finitely generated, and we get an exact sequence of right -modules . Let be any direct system of -torsion-free injective right -modules (with directed). Then, is -FP-injective by (8), so and hence . Thus, we have a commutative diagram:

 with exact rows. Since and are isomorphisms by 25.4(d) in [10], is an isomorphism by the Five Lemma. Now, let be any direct system of -torsion-free modules (with directed). Then, we have a commutative diagram with exact rows:

 where is the injective hull of . Since is finitely generated, by 24.9 in [10], the maps , , and are monic. Since is a hereditary torsion theory and is -torsion-free, by Proposition 3.2, Chap VI in [9], is -torsion-free. And so, by the above proof, is an isomorphism. Hence, is also an isomorphism by the Five Lemma again, and then is -finitely presented by Proposition 2.5 (3) [4], and thus is --presented. Therefore, is right --coherent.  Since is a pure submodule, the pure exact sequence induces a split exact sequence . Since is -torsion-free and -FP-injective, by (9), is -flat, so is also -flat, and thus is -FP-injective by Corollary 2.8 in [2].  Let be a direct system of -torsion-free -FP-injective modules. Then by Proposition 1 in [7], we have a map-pure, and hence pure exact sequence . Observing that is -torsion-free and -FP-injective, by (10), we have that is -FP-injective.

We call a right -module weakly n-FP-injective if for any -presented right -module . Let . Then, we have the following results.

Corollary 2. Let n be a positive integer. Then, the following statements are equivalent for a ring R:

(1)R is right n-coherent(2) for any n-presented right -module A and direct system of right -modules(3) for any n-presented right -module A and each right -module X(4) for any n-presented right -module A and each injective right -module E(5)If X is a weakly n-FP-injective module, then X is n-FP-injective(6)Any direct limit of n-FP-injective modules is n-FP-injective(7)Any direct limit of FP-injective modules is n-FP-injective(8)Any direct limit of injective modules is n-FP-injective(9)A right -module X is n-FP-injective if and only if is n-flat(10)If Y is a pure submodule of an n-FP-injective right -module X, then is n-FP-injective

We note that the equivalences of (1), (2), (6), and (9) in Corollary 2 appeared in Theorem 3.1 in [2].

Corollary 3. Let be a hereditary torsion theory of mod-R and n a positive integer. If R is right -n-coherent, then a -torsion-free module X is n-FP-injective if and only if is n-FP-injective.

Proof.

  Let be a -torsion-free -FP-injective module. Since is right --coherent, by Theorem 2 (9), is -flat, and so is -FP-injective by Proposition 2.3 in [2].  Since is -FP-injective and is a pure submodule of , so, by Proposition 2.6 in [2], X is -FP-injective.

Let ; then, by Theorem 2, we can obtained several results on right -coherent rings.

Corollary 4. Let be a hereditary torsion theory of mod-R. Then, the following statements are equivalent for the ring R:

(1)R is right -coherent(2) for any finitely presented right -module A and direct system of -torsion-free modules(3) for any finitely presented right -module A and each -torsion-free module X(4) for any finitely presented right -module A and each -torsion-free injective module E(5)If X is a -1-FP-injective module, then X is FP-injective(6)Any direct limit of -torsion-free FP-injective modules is FP-injective(7)Any direct limit of -torsion-free injective modules is FP-injective.(8)A -torsion-free module X is FP-injective if and only if is flat(9)If Y is a pure submodule of a -torsion-free FP-injective module X, then is FP-injective

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research was supported by the Natural Science Foundation of Zhejiang Province, China (LY18A010018).

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